Cauchy singular integral operator with parameters in Log-Hölder spaces

This paper is motivated by a claim in the classical textbook of Muskhelishvili concerning the Cauchy singular integral operator S on Hölder functions with parameters. To the contrary of the claim, a counter example was constructed by Tumanov which shows that S with parameters fails to maintain the s...

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Bibliographic Details
Published in:Journal d'analyse mathématique (Jerusalem) 2021-12, Vol.145 (1), p.357-379
Main Authors: Pan, Yifei, Zhang, Yuan
Format: Article
Language:English
Subjects:
Abstract Harmonic Analysis
Analysis
Dynamical Systems and Ergodic Theory
Functional Analysis
Mathematics
Mathematics and Statistics
Operators (mathematics)
Parameters
Partial Differential Equations
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Summary:This paper is motivated by a claim in the classical textbook of Muskhelishvili concerning the Cauchy singular integral operator S on Hölder functions with parameters. To the contrary of the claim, a counter example was constructed by Tumanov which shows that S with parameters fails to maintain the same Hölder regularity with respect to the parameters. In view of the example, the behavior of the Cauchy singular integral operator with parameters between a type of Log-Hölder spaces is investigated to obtain the sharp norm estimates. At the end of the paper, we discuss its application to the ∂ ¯ problem on product domains.
ISSN:0021-7670
1565-8538
DOI:10.1007/s11854-021-0195-y